If there is universal healthcare, caretakers for the elderly and the orphaned should be available. That means a young kid can ask for a caretaker and receive that aid. Kinda like an reverse adoption, where the kid chooses the parent, rather than the other way around.

The government can send a representative to households or schools with a kid under 10 years of age, with the job of asking whether they want to stay. Do this once a year, giving the kid a tablet through which they can securely send a simple survey without showing their parent what they put on it. Depending on what the kid wants, they stay with their family or can tell the state that they are unhappy with where they are.

It wouldn’t be perfect, but at least it gives pathways out of bad situations.